Hannu Korhonen, GeoGebra Ambassador, Nov 22, 2017

The geometry on a sphere is a little bit different as the geometry of our usual environment. Many thanks to Lénárt István and his geometry course in Finland 2003 for understanding these ideas.

A sphere in 3D GeoGebra is defined by the center point and a point on the surface of the sphere.

• 1. Sphere

Points are same kind of objects in every space and every geometry.

• 1. Spherical point

A circle whose center point is in the center point of the sphere is called a [b]great circle[/b]. The shortest path between two points is going along the great circle. So, a great circle is the nearest equivalent of an ordinary line. It would be easy to understand that there are no real lines on the surface of a sphere. Two great lines always cut each other.

• 1. Great circle
• 2. Great circles

There are no parallels in a sphere geometry. So, every pair of great circles are intersecting each others. Then, they are forming an angle, in fact, eight angles.

• 1. Spherical angle

Because there are no real lines on the surface of a sphere, there can not be line segments. The nearest equivalent is a [b]spherical segment[/b], called only a segment in what follows. A segment is the part of a great circle between two points.

• 1. Spherical segment

Two great circles intersect each others always in two opposite points (poles) on the surface of a sphere. And then, a figure with two vertices is boarded by them. In fact, there are four figures of same kind. The vertex angles of the figure are equal. Because the figure has two vertices and two vertex angles only, it could be called a [b]diangle[/b].

• 1. Spherical diangle 1
• 2. Spherical diangle 2

A [b]spherical triangle[/b] has three vertices and three spherical segments as sides, exactly in a same way as a triangle in our ordinary geometry. Change the size by dragging the point P. Because the angles are defining the measure of the area of a spherical triangle, it is often useful to show the sizes in radians. See the second work sheet.

• 1. Spherical triangle
• 2. Spherical triangle 2

A [b]spherical circle[/b] lies on the surface of a sphere. In the spherical geometry, its center and radius are also on the surface. Because there is no special tool for a spherical circle in GeoGebra, the circle must be constructed in an other way. However, the circle behaves like a real spherical circle.

• 1. Spherical circle